Aram Arutyunov

Dmitry Karamzin

F. Lobo Pereira

Repositório Aberto da Universidade do Porto

Necessary Conditions of Optimality for Abnormal Problemswith Equality and Inequality Constraints1Aram Arutyunov?, Dmitry Karamzin†‡, Fernando Pereira‡? PFRU - Peoples Friendship Russian University, Differential Equations and Functional Analysis Dept.,6 Mikluka-Maklai St., 117198 Moscow, Russia, arutun@orc.ru† CCAS - Dorodnicyn Computing Centre of the Russian Academy of Sciences, 40 Vavilova St., 119991Moscow GSP-1, Russia, dmitry karamzin@mail.ru‡ FEUP - Faculdade de Engenharia, Universidade do Porto, Institute for Systems & Robotics - Porto,Dr. Roberto Frias St., 4200-465 Porto, Portugal, flp@fe.up.ptThe aim of this paper is to derive second-order necessary conditions for an abnormal local minimizerof problem (P) which improve the ones presented earlier, [1, 2].We consider the following mathematical programming problem:f(x)→ min, F1(x) = 0, F2(x) ≤ 0. (P).Here, X is a linear space, and f : X → R1, F1 : X → Rk1 , and F2 : X → Rk2 are given mappings (wherek1 and k2 are fixed) assumed to be smooth in a sense to be specified.If x0 is abnormal, i.e. im ∂F1∂x (x0) 6= Rk1 , then it is not difficult to find a simple example revealingthat the usual second-order necessary conditions do not hold in general.Moreover, in [2], meaningful second order necessary conditions for problem (P) were obtained withouta priori normality assumptions of the point x0. In order to formulate these results from [2], let us introducethe Lagrange function L : X ×R1 ×Rk1 ×Rk2 → R1 of problem (P)L(x, λ) = λ0f(x) + 〈λ1, F1(x)〉+ 〈λ2, F2(x)〉,λ = (λ0, λ1, λ2), λ0 ∈ R1, λ1 ∈ Rk1 , λ2 ∈ Rk2 .Let x0 be a local minimizer for problem (P), and the mappings Fi and f be twice continuously differ-entiable. Denote by Λ(x0) the set of all Lagrange multipliers λ = (λ0, λ1, λ2) satisfying the Lagrangemultipliers rule at the point x0:∂L∂x(x0, λ) = 0, λ0 ≥ 0, 〈λ2, F2(x0)〉 = 0, λ2 ≥ 0, |λ| = 1.For the sake of simplicity assume that F2(x0) = 0, and denote by Λa the set of all Lagrange multipliersλ ∈ Λ(x0) for which there exists a linear subspace Π = Π(λ) ⊆ X satisfyingΠ ⊆ ker ∂F1∂x(x0) ∩ ker ∂F2∂x(x0), codimΠ ≤ k1 + k2, ∂2L∂x2(x0, λ)[x, x] ≥ 0, ∀x ∈ Π,where codim means codimension of a linear subspace.In [2], it was proved that, for any vector h ∈ X satisfying ∂F1∂x (x0)h = 0, ∂F2∂x (x0)h ≤ 0, and〈∂f∂x (x0), h〉 ≤ 0, there exists a Lagrange multiplier λ ∈ Λa(x0) (depending on h), such that∂2L∂x2(x0, λ)[h, h] ≥ 0.These necessary conditions are a natural generalization of the classical ones, [3], in the abnormal case.Note that the non-emptiness of the set Λa(x0) is by itself a significant necessary optimality condition.The above mentioned result in [2] was afterwards generalized in [4] (with the help of the techniquedeveloped in [5]) to a problem featuring the more general set-inclusion constraints F (x) ∈ C, where theset C is assumed to be merely closed. On the other hand, the necessary optimality conditions for theproblem with only equality type constraints were, under the additional assumption of abnormality of thepoint x0, strengthened in [6]. More specifically, in this reference, it is shown that if´the local minimizer x0is abnormal, then, the set Λr(x0) in the necessary optimality conditions presented above can be replaced1The research was supported by SFRH/BPD/26231/2006 and RFBR (Russia), projects 08-01-00023, 08-01-00092, 08-01-00450 and FCT (Portugal), projects PTDC/EEA-ACR/75242/2006, and POSI/EEA-SRI/61831/2004.by the smaller set that contains all λ ∈ Λ(x0) such that |λ| = 1 and for which there exists a linearsubspace Π = Π(λ) ⊆ X satisfying:Π ⊆ ker ∂F1∂x(x0), codimΠ ≤ k1 − 1, ∂2L∂x2(x0, λ)[x, x] ≥ 0, ∀x ∈ Π.The main goal of this article is to extend this result to abnormal minimizers of the more generalmathematical programming problem (P). The approach we use is based on a perturbation methoddeveloped in [2] and on methods of real algebraic geometry, [7]In order to state our main result, we consider x0 ∈ X to be the local minimizer in problem (P), andassume that the mappings f , F1 and F2, are twice continuously differentiable in a neighborhood of x0with respect to the finite topology. By this, we mean that, for any finite dimensional subspace M ∈ Mcontaining the point x0, the restrictions of f , F1 and F2 to M are twice continuously differentiable insome (M -dependent) neighborhood of vector x0.We also consider the Lagrange multipliers λ, the Lagrange function L : X×R1×Rk1×Rk2 → R1 andthe set Λ(x0) defined as above. Now, denote by I = I(x0) the set of all indexes i ∈ {1, .., k2} such thatF i2(x0) = 0. (Fis(x) are the coordinates of the vector Fs(x), s = 1, 2). For an integer nonnegative numberr, we denote by Λr(x0) the set of vectors λ ∈ Λ(x0) such that there exists a linear subspace Π = Π(λ)satisfyingΠ(λ) ⊆ ker ∂F1∂x(x0)⋂(⋂i∈Iker∂F i2∂x(x0))such that codimΠ ≤ r, and ∂2L∂x2(x0, λ)[x, x] ≥ 0 ∀x ∈ Π.Consider the cone of critical directions at point x0:K(x0) ={x ∈ X :〈∂f∂x(x0), x〉≤ 0, ∂F1∂x(x0)x = 0,〈∂F i2∂x(x0), x〉≤ 0, i ∈ I}.Put k = k1+ |I(x0)|, where |I| denotes the number of elements in the set I. We shall say that a point x0is abnormal, if k > 0 and the vectors ∂Fj1∂x (x0), j = 1, .., k1,∂F i2∂x (x0), i ∈ I are linearly dependent.Theorem 2.1 Let the point x0 be a local minimizer for problem (P). Assume that x0 is an abnormalpoint. Then, Λk−1(x0) 6= ∅, and the following inequality holds for any vector h ∈ K(x0)maxλ∈Λk−1(x0)∂2L∂x2(x0, λ)[h, h] ≥ 0.This theorem only deals with abnormal minimizers. However, if a minimizer is normal then second-order necessary conditions are well known. See for example [2, 3]. These classical second-order necessaryconditions do not hold for abnormal minimizers, and so, the issues of when do classical second-ordernecessary conditions still follow from our theorem and when it is possible to use an universal Lagrangemultiplier in our main result by omitting the maximum operation arise.References[1] Arutyunov, A.V.: Optimality conditions in abnormal extremal problems. Systems & Control Letters,27 (5), 279–284 (1996).[2] Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Kluwer AcademicPublishers, Dordrecht/Boston/London (2000).[3] Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Consultants Bureau, New York(1987).[4] Arutyunov, A.V., Pereira, F.L.: Second-order necessary optimality conditions for problems without apriori normality assumptions. Mathematics of Operations Research, INFORMS, 31 (1), 1–12 (2006)[5] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I and II, Springer (2006).[6] Arutyunov, A.V., Karamzin, D.Yu.: Necessary Optimality Conditions in an Abnormal OptimizationProblem with Equality Constraints. Computational Mathematics and Mathematical Physics, 46 (8),1293–1298 (2006).[7] Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Springer, A Series of Modern Surveysin Mathematics, (1998).

Conditions of optimality for abnormal problems

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Conditions of optimality for abnormal problems

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